When the 9th of Ab falls on Shabbat, the Tisha B’Ab fast is delayed until Sunday, and on Shabbat, one is allowed to enjoy large, festive meals just as he does on a normal Shabbat. The Sages said that one may enjoy meals on this Shabbat "Ki’se’udat Shelomo Ha’melech Be’sha’ato" – even as large and robust as the meals of King Shlomo.
A number of Poskim addressed the interesting question that arises in the case of a youngster who reaches the age of Misva obligation on Sunday, the 10th of Ab. Must this youngster observe the complete fast, since he is, after all, already a Bar-Misva? Or, is he still considered a minor with respect to this fast, since this fast is observed in place of the fast that was required the previous day, when this youngster was still considered a minor?
This issue depends on the conceptual question as to how we view the observance of Tisha B’Ab on Sunday when the 9th of Ab falls on Shabbat. Do we view the fast as a "make-up" of the fast which could not be observed on Shabbat? Or, did the Sages enact from the outset that when the 9th of Ab falls on Shabbat, the new date of Tisha B’Ab is the 10th of Ab? If Sunday’s fast is a "make-up" of the missed fast, then the boy in this case is not required to observe the compete fast like an adult, since he was not required to do so the previous day. However, if we view Sunday as the day initially instituted as Tisha B’Ab in this circumstance, then this youngster is no different from any other halachic adult, and is required to observe the complete fast.
Some suggest bringing proof to the former perspective from the comments of the Mishna Berura (Rav Yisrael Meir Kagan of Radin, 1839-1933) regarding the question of whether one must limit his festivity during the Shabbat meals when the 9th of Ab falls on Shabbat. The Mishna Berura cites those who claim that although one may eat large meals, these meals should not be conducted in an aura of joy and festivity, but should rather be somber, since, after all, it is Tisha B’Ab. According to others, however, if one normally has festive, joyous meals, and on this Shabbat he conducts himself somberly, this would be a public expression of mourning on Shabbat, which is not appropriate. This entire discussion seems to work on the assumption that although in practice, the fast is observed on Sunday, Shabbat is still treated as Tisha B’Ab, and the question thus becomes as to how this status of the day affects the nature of meals. In other words, the Sages did not "reschedule" Tisha B’Ab on Sunday, but rather required fasting on Sunday since fasting not possible on Shabbat. If so, then we view the Sunday fast as a "make-up" of the fast that was to be observed on Shabbat, such that a youngster who turns 12 or 13 or Sunday would not be required to observe the complete fast.
Nevertheless, Rav Shmuel Wosner (1913-2015), in his Shebet Ha’levi (4:72), writes that in his view, the Sages designated Sunday as the day on which Tisha B’Ab is observed when the 9th of Ab falls on Shabbat, and therefore a youngster in this situation is required to observe the entire fast, just like any other adult.
Summary: If a youngster becomes a Bar-Misva on Sunday, the 10th of Ab – which is observed as Tisha B’Ab – then he must observe the entire fast like an adult, even though he was not yet a Bar-Misva on the 9th of Ab.